Transcript Slide 1

Study of Harmonic oscillator eigenstates
H 
p
2

x
2
2
2
(all parameters = 1)
Since p 
1 
i x
consider a local “coarse grained”
momentum:
p ave  x  

x x
xx


dx
and compare with classical result
p
2E  x
2
Study of Harmonic oscillator eigenstates
H 
p
2

x
2
2
2
(all parameters = 1)
Since p 
1 
i x
consider a local “coarse grained”
momentum:
p ave  x  

x x
xx


dx
and compare with classical result
Note: the actual coarse
graining shown in the
next slides is a very
rough numerical
approximation to the
equation shown here
p
2E  x
2
55th harmonic oscillator eigenstate
140
2
p ave  x2
|d/dx|
2E-x2 (classical)
V
120
100
E
80
60
40
20
0
-10
-5
0
x
5
10
45th harmonic oscillator eigenstate
100
2
p ave  x2
|d/dx|
90
2E-x2 (classical)
V
80
70
E
60
50
40
30
20
10
0
-10
-5
0
x
5
10
35th harmonic oscillator eigenstate
100
2
p ave  x2
|d/dx|
90
2E-x2 (classical)
V
80
70
E
60
50
40
30
20
10
0
-10
-5
0
x
5
10
25th harmonic oscillator eigenstate
70
2
p ave  x2
|d/dx|
2E-x2 (classical)
V
60
50
E
40
30
20
10
0
-10
-5
0
x
5
10
15th harmonic oscillator eigenstate
45
2
p ave  x2
|d/dx|
40
2E-x2 (classical)
V
35
30
E
25
20
15
10
5
0
-6
-4
-2
0
x
2
4
6
5th harmonic oscillator eigenstate
14
2
p ave  x2
|d/dx|
2E-x2 (classical)
V
12
10
E
8
6
4
2
0
-4
-3
-2
-1
0
x
1
2
3
4
0th harmonic oscillator eigenstate
3.5
2
p ave  x2
|d/dx|
2E-x2 (classical)
V
3
2.5
Rough numerical coarse
graining scheme breaking
the symmetry here!
E
2
1.5
1
0.5
0
-1
-0.5
0
x
0.5
1